3.733 \(\int (d+e x)^{-5-2 p} \left (a+c x^2\right )^p \, dx\)

Optimal. Leaf size=436 \[ \frac{c^2 d \left (\sqrt{-a}-\sqrt{c} x\right ) \left (a+c x^2\right )^p (d+e x)^{-2 p-1} \left (3 a e^2-c d^2 (2 p+3)\right ) \left (-\frac{\left (\sqrt{-a}+\sqrt{c} x\right ) \left (\sqrt{-a} e+\sqrt{c} d\right )}{\left (\sqrt{-a}-\sqrt{c} x\right ) \left (\sqrt{c} d-\sqrt{-a} e\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{(2 p+1) (2 p+3) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )^3}+\frac{c e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+1)} \left (a e^2 (2 p+3)-c d^2 \left (2 p^2+8 p+9\right )\right )}{2 (p+1) (p+2) (2 p+3) \left (a e^2+c d^2\right )^3}-\frac{c d e (p+3) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3}}{(p+2) (2 p+3) \left (a e^2+c d^2\right )^2}-\frac{e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )} \]

[Out]

-((c*d*e*(3 + p)*(d + e*x)^(-3 - 2*p)*(a + c*x^2)^(1 + p))/((c*d^2 + a*e^2)^2*(2
+ p)*(3 + 2*p))) + (c*e*(a*e^2*(3 + 2*p) - c*d^2*(9 + 8*p + 2*p^2))*(a + c*x^2)
^(1 + p))/(2*(c*d^2 + a*e^2)^3*(1 + p)*(2 + p)*(3 + 2*p)*(d + e*x)^(2*(1 + p)))
- (e*(a + c*x^2)^(1 + p))/(2*(c*d^2 + a*e^2)*(2 + p)*(d + e*x)^(2*(2 + p))) + (c
^2*d*(3*a*e^2 - c*d^2*(3 + 2*p))*(Sqrt[-a] - Sqrt[c]*x)*(d + e*x)^(-1 - 2*p)*(a
+ c*x^2)^p*Hypergeometric2F1[-1 - 2*p, -p, -2*p, (2*Sqrt[-a]*Sqrt[c]*(d + e*x))/
((Sqrt[c]*d - Sqrt[-a]*e)*(Sqrt[-a] - Sqrt[c]*x))])/((Sqrt[c]*d + Sqrt[-a]*e)*(c
*d^2 + a*e^2)^3*(1 + 2*p)*(3 + 2*p)*(-(((Sqrt[c]*d + Sqrt[-a]*e)*(Sqrt[-a] + Sqr
t[c]*x))/((Sqrt[c]*d - Sqrt[-a]*e)*(Sqrt[-a] - Sqrt[c]*x))))^p)

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Rubi [A]  time = 1.16673, antiderivative size = 436, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ \frac{c^2 d \left (\sqrt{-a}-\sqrt{c} x\right ) \left (a+c x^2\right )^p (d+e x)^{-2 p-1} \left (3 a e^2-c d^2 (2 p+3)\right ) \left (-\frac{\left (\sqrt{-a}+\sqrt{c} x\right ) \left (\sqrt{-a} e+\sqrt{c} d\right )}{\left (\sqrt{-a}-\sqrt{c} x\right ) \left (\sqrt{c} d-\sqrt{-a} e\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{(2 p+1) (2 p+3) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )^3}+\frac{c e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+1)} \left (a e^2 (2 p+3)-c d^2 \left (2 p^2+8 p+9\right )\right )}{2 (p+1) (p+2) (2 p+3) \left (a e^2+c d^2\right )^3}-\frac{c d e (p+3) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3}}{(p+2) (2 p+3) \left (a e^2+c d^2\right )^2}-\frac{e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 (p+2) \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(-5 - 2*p)*(a + c*x^2)^p,x]

[Out]

-((c*d*e*(3 + p)*(d + e*x)^(-3 - 2*p)*(a + c*x^2)^(1 + p))/((c*d^2 + a*e^2)^2*(2
+ p)*(3 + 2*p))) + (c*e*(a*e^2*(3 + 2*p) - c*d^2*(9 + 8*p + 2*p^2))*(a + c*x^2)
^(1 + p))/(2*(c*d^2 + a*e^2)^3*(1 + p)*(2 + p)*(3 + 2*p)*(d + e*x)^(2*(1 + p)))
- (e*(a + c*x^2)^(1 + p))/(2*(c*d^2 + a*e^2)*(2 + p)*(d + e*x)^(2*(2 + p))) + (c
^2*d*(3*a*e^2 - c*d^2*(3 + 2*p))*(Sqrt[-a] - Sqrt[c]*x)*(d + e*x)^(-1 - 2*p)*(a
+ c*x^2)^p*Hypergeometric2F1[-1 - 2*p, -p, -2*p, (2*Sqrt[-a]*Sqrt[c]*(d + e*x))/
((Sqrt[c]*d - Sqrt[-a]*e)*(Sqrt[-a] - Sqrt[c]*x))])/((Sqrt[c]*d + Sqrt[-a]*e)*(c
*d^2 + a*e^2)^3*(1 + 2*p)*(3 + 2*p)*(-(((Sqrt[c]*d + Sqrt[-a]*e)*(Sqrt[-a] + Sqr
t[c]*x))/((Sqrt[c]*d - Sqrt[-a]*e)*(Sqrt[-a] - Sqrt[c]*x))))^p)

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Rubi in Sympy [A]  time = 134.374, size = 384, normalized size = 0.88 \[ \frac{c^{2} d \left (\frac{\left (\sqrt{c} d + e \sqrt{- a}\right ) \left (\sqrt{c} x + \sqrt{- a}\right )}{\left (\sqrt{c} d - e \sqrt{- a}\right ) \left (\sqrt{c} x - \sqrt{- a}\right )}\right )^{- p} \left (a + c x^{2}\right )^{p} \left (d + e x\right )^{- 2 p - 1} \left (- \sqrt{c} x + \sqrt{- a}\right ) \left (3 a e^{2} - 2 c d^{2} p - 3 c d^{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - 2 p - 1, - p \\ - 2 p \end{matrix}\middle |{\frac{2 \sqrt{c} \sqrt{- a} \left (d + e x\right )}{\left (\sqrt{c} d - e \sqrt{- a}\right ) \left (- \sqrt{c} x + \sqrt{- a}\right )}} \right )}}{\left (2 p + 1\right ) \left (2 p + 3\right ) \left (a e^{2} + c d^{2}\right )^{3} \left (\sqrt{c} d + e \sqrt{- a}\right )} - \frac{c d e \left (a + c x^{2}\right )^{p + 1} \left (d + e x\right )^{- 2 p - 3} \left (p + 3\right )}{\left (p + 2\right ) \left (2 p + 3\right ) \left (a e^{2} + c d^{2}\right )^{2}} + \frac{c e \left (a + c x^{2}\right )^{p + 1} \left (d + e x\right )^{- 2 p - 2} \left (- c d^{2} \left (p + 3\right ) + \left (2 p + 3\right ) \left (a e^{2} - c d^{2} \left (p + 2\right )\right )\right )}{2 \left (p + 1\right ) \left (p + 2\right ) \left (2 p + 3\right ) \left (a e^{2} + c d^{2}\right )^{3}} - \frac{e \left (a + c x^{2}\right )^{p + 1} \left (d + e x\right )^{- 2 p - 4}}{2 \left (p + 2\right ) \left (a e^{2} + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(-5-2*p)*(c*x**2+a)**p,x)

[Out]

c**2*d*((sqrt(c)*d + e*sqrt(-a))*(sqrt(c)*x + sqrt(-a))/((sqrt(c)*d - e*sqrt(-a)
)*(sqrt(c)*x - sqrt(-a))))**(-p)*(a + c*x**2)**p*(d + e*x)**(-2*p - 1)*(-sqrt(c)
*x + sqrt(-a))*(3*a*e**2 - 2*c*d**2*p - 3*c*d**2)*hyper((-2*p - 1, -p), (-2*p,),
2*sqrt(c)*sqrt(-a)*(d + e*x)/((sqrt(c)*d - e*sqrt(-a))*(-sqrt(c)*x + sqrt(-a)))
)/((2*p + 1)*(2*p + 3)*(a*e**2 + c*d**2)**3*(sqrt(c)*d + e*sqrt(-a))) - c*d*e*(a
+ c*x**2)**(p + 1)*(d + e*x)**(-2*p - 3)*(p + 3)/((p + 2)*(2*p + 3)*(a*e**2 + c
*d**2)**2) + c*e*(a + c*x**2)**(p + 1)*(d + e*x)**(-2*p - 2)*(-c*d**2*(p + 3) +
(2*p + 3)*(a*e**2 - c*d**2*(p + 2)))/(2*(p + 1)*(p + 2)*(2*p + 3)*(a*e**2 + c*d*
*2)**3) - e*(a + c*x**2)**(p + 1)*(d + e*x)**(-2*p - 4)/(2*(p + 2)*(a*e**2 + c*d
**2))

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Mathematica [B]  time = 127.109, size = 2500, normalized size = 5.73 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(d + e*x)^(-5 - 2*p)*(a + c*x^2)^p,x]

[Out]

-((a + c*x^2)^p*(1 - (d + e*x)/(d + Sqrt[-(a/c)]*e))^(1 + p)*(6*(d + Sqrt[-(a/c)
]*e)^4*p*(Sqrt[-(a/c)] + x)*Gamma[-p]*Gamma[-2*(1 + p)]*Hypergeometric2F1[1, -p,
-3 - 2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))]
+ 22*(d + Sqrt[-(a/c)]*e)^4*p^2*(Sqrt[-(a/c)] + x)*Gamma[-p]*Gamma[-2*(1 + p)]*
Hypergeometric2F1[1, -p, -3 - 2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]
*e)*(Sqrt[-(a/c)] + x))] + 24*(d + Sqrt[-(a/c)]*e)^4*p^3*(Sqrt[-(a/c)] + x)*Gamm
a[-p]*Gamma[-2*(1 + p)]*Hypergeometric2F1[1, -p, -3 - 2*p, (2*Sqrt[-(a/c)]*(d +
e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 8*(d + Sqrt[-(a/c)]*e)^4*p^4*
(Sqrt[-(a/c)] + x)*Gamma[-p]*Gamma[-2*(1 + p)]*Hypergeometric2F1[1, -p, -3 - 2*p
, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 6*(d +
Sqrt[-(a/c)]*e)^2*p*(Sqrt[-(a/c)] + x)*(d + e*x)^2*Gamma[-p]*Gamma[-2*(1 + p)]*
Hypergeometric2F1[1, -p, -1 - 2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]
*e)*(Sqrt[-(a/c)] + x))] + 12*(d + Sqrt[-(a/c)]*e)^2*p^2*(Sqrt[-(a/c)] + x)*(d +
e*x)^2*Gamma[-p]*Gamma[-2*(1 + p)]*Hypergeometric2F1[1, -p, -1 - 2*p, (2*Sqrt[-
(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 6*(d + Sqrt[-(a/c
)]*e)*p*(Sqrt[-(a/c)] + x)*(d + e*x)^3*Gamma[-p]*Gamma[-2*(1 + p)]*Hypergeometri
c2F1[1, -p, -2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)]
+ x))] + 6*(d + Sqrt[-(a/c)]*e)^3*p*(Sqrt[-(a/c)] + x)*(d + e*x)*Gamma[-p]*Gamm
a[-2*(1 + p)]*Hypergeometric2F1[1, -p, -2*(1 + p), (2*Sqrt[-(a/c)]*(d + e*x))/((
d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 18*(d + Sqrt[-(a/c)]*e)^3*p^2*(Sqrt[-
(a/c)] + x)*(d + e*x)*Gamma[-p]*Gamma[-2*(1 + p)]*Hypergeometric2F1[1, -p, -2*(1
+ p), (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 1
2*(d + Sqrt[-(a/c)]*e)^3*p^3*(Sqrt[-(a/c)] + x)*(d + e*x)*Gamma[-p]*Gamma[-2*(1
+ p)]*Hypergeometric2F1[1, -p, -2*(1 + p), (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt
[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 18*Sqrt[-(a/c)]*(d + Sqrt[-(a/c)]*e)^2*p*(d +
e*x)^2*Gamma[-3 - 2*p]*Gamma[1 - p]*Hypergeometric2F1[2, 1 - p, -1 - 2*p, (2*Sq
rt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 48*Sqrt[-(a/c
)]*(d + Sqrt[-(a/c)]*e)^2*p^2*(d + e*x)^2*Gamma[-3 - 2*p]*Gamma[1 - p]*Hypergeom
etric2F1[2, 1 - p, -1 - 2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(S
qrt[-(a/c)] + x))] + 24*Sqrt[-(a/c)]*(d + Sqrt[-(a/c)]*e)^2*p^3*(d + e*x)^2*Gamm
a[-3 - 2*p]*Gamma[1 - p]*Hypergeometric2F1[2, 1 - p, -1 - 2*p, (2*Sqrt[-(a/c)]*(
d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 33*Sqrt[-(a/c)]*(d + e*x)
^4*Gamma[-3 - 2*p]*Gamma[1 - p]*Hypergeometric2F1[2, 1 - p, 1 - 2*p, (2*Sqrt[-(a
/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 22*Sqrt[-(a/c)]*p*(
d + e*x)^4*Gamma[-3 - 2*p]*Gamma[1 - p]*Hypergeometric2F1[2, 1 - p, 1 - 2*p, (2*
Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 54*Sqrt[-(a
/c)]*(d + Sqrt[-(a/c)]*e)*p*(d + e*x)^3*Gamma[-3 - 2*p]*Gamma[1 - p]*Hypergeomet
ric2F1[2, 1 - p, -2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(
a/c)] + x))] + 36*Sqrt[-(a/c)]*(d + Sqrt[-(a/c)]*e)*p^2*(d + e*x)^3*Gamma[-3 - 2
*p]*Gamma[1 - p]*Hypergeometric2F1[2, 1 - p, -2*p, (2*Sqrt[-(a/c)]*(d + e*x))/((
d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 18*Sqrt[-(a/c)]*(d + e*x)^4*Gamma[-3
- 2*p]*Gamma[1 - p]*HypergeometricPFQ[{2, 2, 1 - p}, {1, 1 - 2*p}, (2*Sqrt[-(a/c
)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 12*Sqrt[-(a/c)]*p*(d
+ e*x)^4*Gamma[-3 - 2*p]*Gamma[1 - p]*HypergeometricPFQ[{2, 2, 1 - p}, {1, 1 - 2
*p}, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 18*
Sqrt[-(a/c)]*(d + Sqrt[-(a/c)]*e)*p*(d + e*x)^3*Gamma[-3 - 2*p]*Gamma[1 - p]*Hyp
ergeometricPFQ[{2, 2, 1 - p}, {1, -2*p}, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-
(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 12*Sqrt[-(a/c)]*(d + Sqrt[-(a/c)]*e)*p^2*(d + e
*x)^3*Gamma[-3 - 2*p]*Gamma[1 - p]*HypergeometricPFQ[{2, 2, 1 - p}, {1, -2*p}, (
2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c)] + x))] + 3*Sqrt[-(
a/c)]*(d + e*x)^4*Gamma[-3 - 2*p]*Gamma[1 - p]*HypergeometricPFQ[{2, 2, 2, 1 - p
}, {1, 1, 1 - 2*p}, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt[-(a/c)]*e)*(Sqrt[-(a/c
)] + x))] + 2*Sqrt[-(a/c)]*p*(d + e*x)^4*Gamma[-3 - 2*p]*Gamma[1 - p]*Hypergeome
tricPFQ[{2, 2, 2, 1 - p}, {1, 1, 1 - 2*p}, (2*Sqrt[-(a/c)]*(d + e*x))/((d + Sqrt
[-(a/c)]*e)*(Sqrt[-(a/c)] + x))]))/(4*e*(d + Sqrt[-(a/c)]*e)^4*p*(1 + p)*(2 + p)
*(1 + 2*p)*(3 + 2*p)*((e*(Sqrt[-(a/c)] - x))/(d + Sqrt[-(a/c)]*e))^p*(Sqrt[-(a/c
)] + x)*(d + e*x)^(2*(2 + p))*Gamma[-p]*Gamma[-2*(1 + p)])

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Maple [F]  time = 0.125, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{-5-2\,p} \left ( c{x}^{2}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(-5-2*p)*(c*x^2+a)^p,x)

[Out]

int((e*x+d)^(-5-2*p)*(c*x^2+a)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 5}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^p*(e*x + d)^(-2*p - 5),x, algorithm="maxima")

[Out]

integrate((c*x^2 + a)^p*(e*x + d)^(-2*p - 5), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c x^{2} + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 5}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^p*(e*x + d)^(-2*p - 5),x, algorithm="fricas")

[Out]

integral((c*x^2 + a)^p*(e*x + d)^(-2*p - 5), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(-5-2*p)*(c*x**2+a)**p,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 5}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^p*(e*x + d)^(-2*p - 5),x, algorithm="giac")

[Out]

integrate((c*x^2 + a)^p*(e*x + d)^(-2*p - 5), x)